Quantum Mechanics
• Erwin Schrödinger developed a
mathematical solution where the e- is
described as a wave.
• Mathematically derived the probability
of finding an e- in a given region of
space. This is known as the ‘electron
density’. The electrons are here 90% of
the time.
• The region of space where the electron
(of a given energy) is most probably
located is known as ‘orbital’.
Quantum Mechanics
• The wave equation is designated
with a lower case Greek psi (ψ).
• The square of the wave equation,
ψ2, gives a probability density
map of where an electron has a
certain statistical likelihood of
being at any given instant in time.
• From the ‘uncertainty principle’
we cannot determine the location
of an e- of any given energy.
The ‘quantum numbers’ n, l, ml are derived from Schrodinger’s
eqn.
•n, the Principal Quantum Number = 1, 2, 3, …infinity
Quantum Numbers
• Solving the wave equation gives a set of
wave functions, or orbitals, and their
corresponding energies.
• Each orbital describes a spatial distribution
of electron density.
• An orbital is described by a set of three
quantum numbers (n, l, ml)
Gives the primary electron shell for the electron.
Analogous to Bohr’s value of n. En = –ℜhc/n2.
(‘bigger’ n value means ‘bigger’ orbitals, and ‘higher’
energy)
•l, the Angular Momentum Quantum Number = 0, 1, 2, …
n-1 This quantum number defines the shape of the orbital.
Gives the electron subshell or orbital the electron can be
found in. Allowed values of ‘l’ are 0 to (n-1).
Value of l
Type of orbital
0
s
1
p
2
d
3
f
•Bigger ‘l’ higher the energy of the orbital
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In summary….
Magnetic Quantum Number, ml
• Describes the three-dimensional orientation of the orbital.
• Values are integers ranging from -l to l:
l
possible values of ml
# orbitals in this subshell
0
0
1
1
-1, 0, +1
3
2
-2, -1, 0, +1, +2
5
3
-3, -2, -1, 0, +1, +2, +3
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Spin Quantum Number, ms
• In the 1920s, it was discovered
that two electrons in the same
orbital do not have exactly the
same energy.
• The “spin” of an electron
describes its magnetic field,
which affects its energy.
• The spin quantum number, ms
has values of +1/2 and –1/2
Electron spin.MOV
Observing a graph of probabilities of finding an electron versus distance from
the nucleus, we see that s orbitals possess n−1 nodes, or regions where there
is 0 probability of finding an electron. ‘s’ orbitals have a zero ‘l ‘ value.
2
p orbitals
•Has an imaginary plane ( nodal surface) that divides the region of
electron density in half. Zero probability of finding an e- here.
•Occurs when l = 1 (dumb-bell shaped orbitals, one nodal surface)
d orbitals
•Occurs when l = 2, orbitals have two nodal surfaces, four regions
Of e- density. Five orbitals, corresponding to ml quantum number.
(dxy, dxz, dyz, dx2-y2, dz2)
•When l = 1, ml values are –1, 0, +1 (corresponds to px, py and pz)
RadialElectronDistribution.MOV
Quantum e cloud.MOV
Orbital Energy Levels of Hydrogen
For the Hydrogen atom, each
Subshell with the same ‘n’ has
same energy.
We say that the orbitals are
degenerate.
For n =2, 2s and 2p orbitals are
degenerate (possess the same
energy).
Energy levels in a non-Hydrogen atom
•As the number of electrons
increases, though, so does
the repulsion between
them.
•Therefore, in manyelectron atoms, orbitals on
the same energy level are
no longer degenerate
* Recall at n = , E = 0
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Pauli Exclusion Principle
• two electrons cannot share the
same set of quantum numbers
within the same system, i.e no
two electrons in the same
atom can have identical sets of
quantum numbers.
(n, l, ml and ms)
•Example: An electron in a 3d orbital might have the
following four quantum numbers, n = 3, l = 2, ml = 0, ms =
+1/2.
A second electron in the same 3d orbital would have the
following quantum numbers, n = 3, l = 2, ml = 0, ms = -1/2.
Electron Configurations
Number denotes the energy level, n=4
Denotes the subshell ‘p’ orbital
Denotes number of e- in the orbital
Hund’s Rule
Orbital Notations
Every orbital in a subshell is singly occupied with one e- before any one
orbital is doubly occupied.
•Each box denotes an orbital
•The half arrows denote an e-
All electrons in singly occupied orbitals have the same spin.
This configuration helps attain the lowest energy configuration within
degenerate orbitals
•The up and down arrows denote
differences of e- spin
Which element is this?
1s22s22p4
ElectronConfigurations.MOV
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Writing e- configurations…
Electronic Configurations of some elements…..
1s
Anomalies..
2px
2py
2pz Full and Condensed spdf Notation
H
↑
1s1
He
↑↓
1s2
Li
↑↓
↑
1s22s1 or [He]2s1
Be
↑↓
↑↓
1s22s2 or [He]2s2
B
↑↓
↑↓
↑
C
↑↓
↑↓
↑
1s22s22p1 or [He]2s22p1
↑
↑
O ↑↓ ↑↓ ↑↓ ↑
↑
F ↑↓ ↑↓ ↑↓ ↑↓ ↑
Ne ↑↓ ↑↓ ↑↓ ↑↓ ↑↓
N
To write spdf notation, read the periodic table from left to
right, row by row, and fill in with e- starting from the
lowest energy orbital available! (this is Aufbau
principle)
2s
↑↓
↑↓
↑
↑
1s22s22p2 or [He]2s22p2
1s22s22p3 or [He]2s22p3
1s22s22p4 or [He]2s22p4
1s22s22p5 or [He]2s22p5
1s22s22p6 or [Ne]
Exceptions to the rule
•Some exceptions to Hund’s rule occur for the transition elements,as
4s and 3d orbitals are very close in energy.
•Occurs in f block elements as well
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Exceptions to the rule
A d subshell that is half-filled or full (ie 5 or 10 electrons) is more
stable than the s subshell of the next shell.
This is because it takes less energy to maintain an electron in a half-filled
d subshell than a filled s subshell. For instance, copper (atomic number
29) has a configuration of [Ar]4s1 3d10 NOT [Ar]4s2 3d9 as one would
expect.
Chromium (atomic number 24) has a configuration of [Ar]4s1 3d5, not
[Ar]4s2 3d4. [Ar] : configuration for argon.
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